In this section we propose a short description on standard inflation
formalism to enable readers to understand the meanings of terms here
involved. Inflation is the period of the early universe that undergoes an
accelerating phase [1]}. This period is equivalent to \(\ddot{a}\)  \(\succ 0.\)  The candidate that can yield this acceleration phase and
responsible for driving inflation is a scalar field named inflaton field.
The pressure and energy of the inflaton field are given by \(p_{\phi }=\frac{1}{2}\dot{\phi }^{2}-V(\phi )\)  and \(\rho _{\phi }=\frac{1}{2}\dot{\phi }^{2}+V(\phi )\) , where \(V(\phi )\)  is the scalar potential. The Friedmann
equation reads \(H^{2}=\frac{8\pi G}{3}\left[ \frac{1}{2}\dot{\phi }^{2}+V(\phi )\right] .\)  During inflation, the potential \(V(\phi )=V\)  depends
only on the inflaton field \(\phi \) . It is supposed that the field equation \(\ddot{\phi }+3H\dot{\phi }=-V^{\prime }(\phi )\) , is well-approximated by \(3H\dot{\phi }=-V^{\prime }(\phi ).\)  The condition for acceleration requires
that \(\dot{\phi }^{2}\prec V,\)  This is called the slow-roll approximation.
With very small \(\dot{\phi }^{2}\) , the Friedmann equation is approximately \(3M_{p}^{2}H^{2}\simeq V,\)  and the flatness conditions \(\varepsilon \)  \(\prec \prec 1\)  and \(\mid \eta \mid \prec \prec 1,\)  where \(\varepsilon =\frac{1}{2}M_{p}^{2}\left( \frac{V^{\prime }}{V}\right) ^{2}\)  and \(\eta =M_{p}^{2}\frac{V^{\prime \prime }}{V^{\prime \prime }}.\)  The amount of inflation can be
measured in term of the e-folding number \(N\)  given by equation \(N=M_{p}^{-2}\int _{\phi _{end}}^{\phi _{\ast }}\left( \frac{V}{V^{\prime }}\right) d\phi ,\)  where \(\phi _{\ast }\)  and \(\phi _{end}\)  are the values of
the scalar field at the epoch when the cosmological scales exit the horizon
and at the end of inflation, respectively.
